||The dependence of thickness and concentration product (TCP) of single-wall carbon nanotubes saturable absorber (SWCNTs SA) on stabilizing and shortening pulse width in passively mode-locked erbium-doped fiber ring laser (MLEDFL) was investigated and measured. The TCP represented the amounts of SWCNTs, which the optical beam encountered when passing through the SWCNTs SA. If the TCP was smaller than 8.25 (μm x wt%), the spectral bandwidth was below 2 nm. The pulse shaping was dominated by its own self amplitude modulation (SAM) of SWCNTs SA. With further increasing TCP, the soliton-like ML operation was achieved and the spectral bandwidth was expanded to 6 nm. For soliton-like mode locking (ML) operation, the area theorem dominated the pulse shaping.|
Through area theorem analysis, the estimation of SPM increased as the TCP increased. The adequate enhanced SPM for balancing the slight negative GVD was provided to generate soliton-like ML pulses shorten the pulse width. However, as the TCP increased, the soliton pulse energy decreased. The decreasing soliton pulse energy restricted the further pulse shortening. The results showed that the dependence of the pulse energy and nonlinear self phase modulation (SPM) on TCP enabled to determine the shortest pulse width in MLEDFL based on the area theorem. At optimized TCP of 70.93 (μm x wt%), it was found that the shortest pulse width of 418 fs.
In addition, based on the estimated SPM from area theorem, the nonlinear refractive index n2 was calculated at the level of 0.4 - 1 x 10^-15 m^2/W that was close to the literature values of 10^-15 - 10^-16 m2/W. It provides another way to estimate the nonlinear refractive index except for the Z-scan measurement. We could also estimate the SPM if an active Z-scan measurement was taken to obtain the nonlinear refractive index of the sample. We realized the trend of pulse energy through few samples in MLEDFL, the behavior of pulse width could be theoretically simulated based on area theorem. Hence, with the area theorem analysis, the optimized TCP of SWCNTs SA could be simulated and estimated to generate the shortest pulse width from the trends of pulse energy and estimated SPM. The significant effect of TCP on pulse energy, SPM, pulse width, and spectral bandwidth of MLFLs suggests that the TCP represents the total amount of SWCNTs in SA, which can be used as one of key parameters for characterizing the passive MLFL pulse width. Through the study of the dependence of TCP on ML pulses in MLEDFL, it may provide a guideline to fabricate an effective SWCNTs SA to generate the shortest pulse width of the MLEDFL.